KURTOSIS

Purpose:

Calculates the kurtosis of a series.

Syntax:

KURTOSIS(series, bias, excess, vardef)

series

-

A series. The z value.

bias

-

Optional. An integer, the bias calculation. Valid methods are:

0:

compute unbiased estimate (default)

1:

compute biased estimate

excess

-

Optional. A real. Excess normalization subtracted from the base calculation. Defaults to 3.0.

vardef

-

Optional. An integer, the variance normalization method.

0:

1 / (N-1)

1:

1 / N

where N is the length of the input series. Defaults to 1/(N-1) for the unbiased computation and 1/N for the biased computation.

Returns:

A scalar, the kurtosis of the series.

Example:

W1: {3, 4, 5, 2, 3, 4, 5, 6, 4, 7}

W2: {kurtosis(w1)}

W3: {kurtosis(w1, 1)}

 

W2 == {-0.151800}, the unbiased or population kurtosis.

W3 == {-0.631321}, the biased kurtosis.

Remarks:

The fourth standardized moment or raw kurtosis is defined as:

 

image\kurtosis01.gif

 

where μ4 is the fourth moment about the mean and σ is the standard deviation.

 

The excess kurtosis is defined as:

 

image\kurtosis02.gif

 

The value 3 is subtracted from the raw value to set the kurtosis of a normal distribution to 0.

 

The biased or sample kurtosis is computed as:

 

image\kurtosis04.gif

 

where the arithmetic mean is defined as:

 

image\std02.gif

 

The unbiased or population kurtosis is computed as:

 

image\kurtosis03.gif

 

By default, KURTOSIS computes the unbiased or population kurtosis as used in SAS, SPSS, and Excel. To calculate the biased kurtosis, set the bias flag to 1.

 

Set excess to 0.0 to compute the raw kurtosis.

 

Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

 

A distribution with a high peak is called leptokurtic, a flat-topped curve is called platykurtic, and the normal distribution is called mesokurtic.

 

For multi-column data, KURTOSIS returns the kurtosis of each column.

See Also:

MEAN

SKEW

STD